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<p>We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies Riemann Hypothesis or Generalized Riemann Hypothesis for Dirichlet <i>L</i>-functions. While such sign constancy is challenging to establish individually, we prove that the summatory functions under study have constant signs on average.</p>

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On variants of Chebyshev’s conjecture

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月10日

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  • Masatoshi Suzuki

摘要

We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies Riemann Hypothesis or Generalized Riemann Hypothesis for Dirichlet L-functions. While such sign constancy is challenging to establish individually, we prove that the summatory functions under study have constant signs on average.